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Quasi-Monte Carlo methods for the Heston model. Registered: Jan Baldeaux. In this paper, we discuss the application of quasi-Monte Carlo methods to the Heston model. We base our algorithms on the Broadie-Kaya algorithm, an exact simulation scheme for the Heston model.


As the joint transition densities are not available in closed-form, the Linear Transformation method due to Imai and Tan, a popular and widely applicable method to improve the effectiveness of quasi-Monte Carlo methods, cannot be employed in the context of path-dependent options when the underlying price process follows the Heston model.

Consequently, we tailor quasi-Monte Carlo methods directly to the Heston model. The contributions of the paper are threefold: We firstly show how to apply quasi-Monte Carlo methods in the context of the Heston model and the SVJ model, secondly that quasi-Monte Carlo methods improve on Monte Carlo methods, and thirdly how to improve the effectiveness of quasi-Monte Carlo methods by using bridge constructions tailored to the Heston and SVJ models. Handle: RePEc:arx:papers A number of possible theoretical explanations have been advanced.

Pdf Monte Carlo And Quasi Monte Carlo Methods 2010 2012

This has been a very research rich area leading to powerful new concepts but a definite answer has not been obtained. A possible explanation of why QMC is good for finance is the following. Consider a tranche of the CMO mentioned earlier.

33. Monte Carlo Methods 2

The integral gives expected future cash flows from a basket of year mortgages at monthly intervals. Because of the discounted value of money variables representing future times are increasingly less important. In a seminal paper I. Sloan and H. In these spaces the dependence on the successive variables can be moderated by weights. If the weights decrease sufficiently rapidly the curse of dimensionality is broken even with a worst case guarantee. This paper led to a great amount of work on the tractability of integration and other problems.

On the other hand, effective dimension was proposed by Caflisch, Morokoff and Owen [23] as an indicator of the difficulty of high-dimensional integration. The purpose was to explain the remarkable success of quasi-Monte Carlo QMC in approximating the very-high-dimensional integrals in finance.

The impact of the arguments of Caflisch et al.

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A number of papers deal with the relationship between the error of QMC and the effective dimension [24]. It is known that QMC fails for certain functions that have high effective dimension. QMC can also be superior to MC and to other methods for isotropic problems, that is, problems where all variables are equally important.

For example, Papageorgiou and Traub [27] reported test results on the model integration problems suggested by the physicist B. Keister [28]. Its error was. These are empirical results. In another theoretical investigation Papageorgiou [30] presented sufficient conditions for fast QMC convergence. The conditions apply to isotropic and non-isotropic problems and, in particular, to a number of problems in computational finance.

He presented classes of functions where even in the worst case the convergence rate of QMC is of order. But this is only a sufficient condition and leaves open the major question we pose in the next section. From Wikipedia, the free encyclopedia. Financial Economics, 4, Portfolio Management, 22 1 , F and Werschulz, A.

Monte Carlo and Quasi-Monte Carlo Methods 2012

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Pliska and M. Dempster eds. Fang and F. Hickernell eds.

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